Compactness and Semicontinuity in Minimax Interchange

Why Topology Matters in Minimax Problems

In a zero-sum game, a minimizing player chooses \(x\) from a strategy set \(X\), a maximizing player chooses \(y\) from \(Y\), and the payoff to the maximizer is \(\Phi(x,y)\). The minimax comparison asks whether \(\inf_{x\in X}\sup_{y\in Y}\Phi(x,y)\) equals \(\sup_{y\in Y}\inf_{x\in X}\Phi(x,y)\). Convexity and concavity help establish that equality, but they are not the whole story. Compactness and semicontinuity determine whether the relevant infima and suprema are actually achieved.

This distinction matters in statistical decision problems. A minimax value can describe the best guaranteed risk even when no strategy achieves that value. Compactness and the appropriate semicontinuity rule out this particular failure on the compact side of a minimax argument. They do not, by themselves, prove minimax interchange.

Semicontinuity: The Level-Set Conditions

Let \(S\) be a topological space and let \(\varphi:S\to\mathbb{R}\). Lower and upper semicontinuity are one-sided forms of continuity: lower semicontinuity permits an upward jump at a limit point, while upper semicontinuity permits a downward jump. The open-level-set characterizations fix which direction is which.

In a metric space these definitions also have sequential forms. Lower semicontinuity means that for every sequence \(s_k\to s\), \(\varphi(s)\leq\liminf_{k\to\infty}\varphi(s_k)\); upper semicontinuity means \(\varphi(s)\geq\limsup_{k\to\infty}\varphi(s_k)\). The level-set definitions apply on general topological spaces, where sequences alone need not capture the topology.

Compactness Gives Attainment

The key attainment result is often called the extreme value theorem for lower semicontinuous functions. Unlike the continuous extreme value theorem, it guarantees a minimum but not necessarily a maximum. The upper semicontinuous counterpart guarantees a maximum.

Proof for the minimum. For each \(s\in S\), the set \(U_s=\{t:\varphi(t)>\varphi(s)-1\}\) is an open neighborhood of \(s\). These neighborhoods cover \(S\). Compactness gives a finite subcover \(U_{s_1},\ldots,U_{s_m}\). For any \(t\in S\), membership in at least one of these sets gives \(\varphi(t)>\varphi(s_i)-1\) for some \(i\). Therefore \(\varphi\) is bounded below by the finite number \(b=\min_i(\varphi(s_i)-1)\). Its infimum \(q\) is consequently finite, since \(\varphi\) is real-valued and \(S\) is nonempty.

For each positive integer \(k\), let \(F_k=\{s:\varphi(s)\leq q+1/k\}\). Lower semicontinuity makes \(F_k\) closed. The definition of infimum makes each \(F_k\) nonempty, and \(F_{k+1}\subseteq F_k\). These nested nonempty closed subsets of the compact space \(S\) have nonempty intersection. If \(s^*\) belongs to that intersection, then \(\varphi(s^*)\leq q+1/k\) for every \(k\), so \(\varphi(s^*)\leq q\). Since \(q\) is the infimum, equality holds: \(\varphi(s^*)=q\).

For the maximum, apply the minimum result to \(-\varphi\). If \(\varphi\) is upper semicontinuous, then \(-\varphi\) is lower semicontinuous; minimizing \(-\varphi\) is equivalent to maximizing \(\varphi\). \(\square\)

How Semicontinuity Enters Minimax Interchange

For a payoff \(\Phi:X\times Y\to\mathbb{R}\), define the two one-variable value functions \(H(x)=\sup_{y\in Y}\Phi(x,y)\) and \(G(y)=\inf_{x\in X}\Phi(x,y)\). The minimax quantities are \(\inf_{x\in X}H(x)\) and \(\sup_{y\in Y}G(y)\). The following preservation facts explain how semicontinuity transfers from the payoff sections to these value functions.

Proof. For any real \(a\), \(\{x:H(x)>a\}=\bigcup_{y\in Y}\{x:\Phi(x,y)>a\}\). Each set in this union is open by lower semicontinuity of the corresponding section, so the union is open. This proves that \(H\) is lower semicontinuous. Similarly, \(\{y:G(y)<a\}=\bigcup_{x\in X}\{y:\Phi(x,y)<a\}\). Each set is open by upper semicontinuity of the section, so \(G\) is upper semicontinuous. \(\square\)

If \(X\) is compact, the attainment theorem applies to \(H\), provided \(H\) is real-valued. In particular, bounded payoffs ensure this condition. Thus the minimizing player attains \(\min_x\sup_y\Phi(x,y)\). If \(Y\) is also compact, the same theorem applied to \(G\) shows that the maximizing player attains \(\max_y\inf_x\Phi(x,y)\). Compactness of only the minimizing strategy set does not, in general, guarantee attainment of the maximizing player’s supremum.

Here is the relevant form of Sion’s minimax theorem, which combines these topological conditions with the convexity and concavity requirements. A function is quasiconvex if each of its sublevel sets is convex, and quasiconcave if each of its superlevel sets is convex.

The theorem’s compactness hypothesis is on \(X\), the minimizing player’s set. Lower semicontinuity in \(x\) is the condition that supports attainment of the left-hand minimum. The right-hand supremum need not be attained unless additional conditions, such as compactness of \(Y\), are available. Semicontinuity and compactness provide the topological part of the argument; the convexity and concavity hypotheses are also essential to Sion’s interchange result.

When Equality Does Not Give an Attained Strategy

If both strategy sets are compact and the relevant value functions have the semicontinuity just established, minimax equality also yields an attained saddle point. To see why, let \(x^*\) minimize \(H\) and \(y^*\) maximize \(G\). If the common minimax value is \(V\), then \(\Phi(x^*,y)\leq H(x^*)=V\) for every \(y\), while \(\Phi(x,y^*)\geq G(y^*)=V\) for every \(x\). In particular, using \(x=x^*\) and \(y=y^*\) shows that \(\Phi(x^*,y^*)=V\). Thus \(\Phi(x^*,y)\leq\Phi(x^*,y^*)\leq\Phi(x,y^*)\) for all \(x,y\).

Common Mistakes and Why These Conditions Matter

• Reversing the open-set tests. Lower semicontinuity means open strict superlevel sets \(\{\varphi>a\}\); upper semicontinuity means open strict sublevel sets \(\{\varphi<a\}\). Reversing them swaps the conditions needed for minima and maxima.
• Assuming compactness alone gives an extremum. Compactness needs the matching regularity: lsc for a minimum, usc for a maximum. An arbitrary discontinuous function on a compact set need not attain either.
• Claiming that minimax equality supplies optimizers. Interchange is a statement about values. To conclude that a player has an optimal strategy, check attainment separately; in particular, the maximizing set may not be compact.
• Using Sion’s theorem without its shape conditions. The required semicontinuity does not replace convexity of the strategy sets and quasiconvexity or quasiconcavity of the payoff sections.

These distinctions are useful when translating a statistical decision problem into a game. A compact set of decision rules, together with lower semicontinuity of worst-case risk in the rule, can justify existence of a minimax rule. The corresponding regularity and compactness on the nature or parameter side determine whether an optimizing opponent strategy exists as well. In either case, verify the topology and the direction of semicontinuity before invoking an attainment or minimax theorem.